On Saturday, I discussed Lewindowsky’s No Lew. We don’t need that level of flood defenses. In that post, I presented a qualitative discussion of a Lewandowsky claim; today, I’m going to show some “math”. The purpose is to explain why the italicized claim in Pancost and Lewandowsky below is exaggerated to the point of being wrong:

Second, the uncertainty in our projections makes adaptation to climate change more expensive and challenging. Suppose we need to build flood defences for a coastal English town. If we could forecast a 1m sea level rise by 2100 without any uncertainty, the town could confidently build flood barriers 1m higher than they are today. However, although sea levels are most likely to rise by about 1m, we’re really looking at a range between 0.3m and 1.7m. Therefore, flood defences must be at least 1.7m higher than today – 70cm higher than they could be in the absence of uncertainty.
And as uncertainty increases, so does the required height of flood defences for non-negotiable mathematical reasons.

(In their blog post, the claim about ‘non-negotiable mathematical reasons’ was linked to a paywall verion their paper. I’ve replaced the link to send you to a free copy.)
Getting mathematical, I will also show that while the final claim “And as uncertainty increases, so does the required height of flood defences for non-negotiable mathematical reasons”, is true, the uncertainty increase can be so small as to be lost when one rounds the required height to the number of significant figures in the estimated sea level change or its stated uncertainty. This makes that claim rather less impressive than it might sound.

Friday, my comment about this was

The idea that “flood defences must be at least 1.7m higher than today” is utter nonsense. The more correct claim is “we need to consider the possibility that flood defenses in 2100 may need to be 1.7 m higher than today”.

Note in the previous discussion, 1.7 m is the upper range of of the previously discussed uncertainty window for sea level rise (1.7m). The general point would appear to be one ‘must’ build to protect to the ‘upper range’– that’s bunk.

Today, I’m going to use a simple “Toy” model to show that, in fact, ‘uncertainty’ does not require anyone to build a wall to math the upper range of projections now or ever. One only needs to build to that range if that sea level rise materializes?

Why don’t we need to do this: Because we can wait.

Of course, I already said this, but today, I’m going to do math. The math will be similar to that done by Lewindowsky and co-authors in the paper he linked. Unlike Lewindowsky and Pancost, I’m not going to suggest that my results are based on ‘non-negotiable’ math. They are based on

1. A decision making algorithm policy makers could chose to follow if they wished.
2. Some simple “toy” assumption permitting a mathematical function to ‘project’ the expected value of sea level rise (slr) and its uncertainty as a function of time. This model will require specifying two parameters.
3. A mathematical model to predict inundation risk. This model will require specifying a parameter. λ
4. Mathematical manipulations themselves.

Though it might not be obvious based on Lew’s verbiage in his recent blog post, Lew did three of the four in his paper. I’ll discuss his assumptions and my extention below. :

1. In his peer reviewed paper, Lew ‘analysis’ is based on the assumption that decision makers build defenses one time and one time only and they act based on current, or at least reasonably current, projections for sea level height in 2100. That is: they don’t get to build a little now and then re-visit needs later.

   In contrast, I will assume that a board deciding on implementing engineered protections that might be required 86 years from now (2014-2100), can elect to build protections in increments. So, for example, they can chose to build in 2 increments, starting by building protections they anticipate will protect through the first 43 years and deferring the decision for further building to the future.

   Note: this assumption is not ‘math’; it will turn out to make a very large difference to our estimate of the height of engineered protections ‘required’ in light of uncertainty in projections.

2. Because of the assumption the board only acts once, Lew didn’t need a toy model to project sea level rise. This simplifies his “math”. You get to decide if it simplified it a bit too much.

   Since I am permitting the board to make prudent decisions, I will assume

   (a) The expected value of sea level rise $latex E[SLR] $ varies quadratically with time and the board will continue to believe this is the functional form. The current level is 0 m; the value in 2100 is 0.5m. It’s currently 2014, they currently believe the SLR varies quadratically with time measured from 2014 reaching 0.5m in 2100. $latex E[SLR]= (0.5m ) (t/86)^{2} $ where time is measured from 2014. (Note: I introduced a parameter here. It’s the exponent of ‘2’.)

   The expected value of sea level rise is illustrated with the black squares in the figure below which shows the current sea level and projections at the end of each build time if we assume 5 build periods:

   
   Note the final expected value for sea level rise is 0.5 m and matches that used by Lewandowsky. His analysis didn’t need additional detail because the board was assumed to lack any critical thinking skills and could act only once.

   Those with critical thinking skills know the entire point of deferring decisions is to permit the current and future boards to use future events to guide their future decisions. The current board can’t know the future value of $latex SLR $, nor can they know the future projections. For the purpose of decision making, they need a model. I’ll suggest this simple one:

   The current board will assume that when some future year Y arrives (e.g. Y=2050), scientists will observe the sea level rise, which we will call “Observed”, $latex O[SLR](Y) $. At that future time, the current board assumes the future board will continue to believe the sea level rise varies quadratically), passing through the pairs $latex (2014, 0) $ and $latex (2050,O(SLR)(Y) $ with minimum at 2014. That future board will be assumed use the same risk methodology the current board likes, but base flood risk calculations for events further into the future on new projections which have been updated based on new observations.

   Note that under this “delayed action” plan, if the observed rate of rising exceeds the current best estimate of the rate of rise, the future board will expect the sea level rate to rise at a faster rate than currently projected; at that point they can build the wall higher than the current board’s best estimate of the build height in the future year. The converse is also true.

   (b) The current board will also assume uncertainty in projected sea level rise also varies quadratically with time measured from the date on which they are making decisions. The current level is 0m. The value in 2100 is 0.36 m. So, in 2014, the uncertainty in projected sea level rise varies as $latex \sigma = (0.36 m) (dt/86)^{2} $ where $latex dt = t-2014 $; note $latex T=2100-2014 = 86 $ in the denominator is the full time period the board is considering when developing their response. (Note: I’ve introduced my second prameter, it’s 2.) The current uncertainty intervals are illustrated by the range bars in the figure above.

   When the future arrives, the current board assumes scientists ability ability to predict sea level rise has not have improved, and that $latex (0.36 m) (dt/86)^{2} $ where dt is now computed using the current year, i.e. dt = t-Year_{current} $; . So, for example, when making decision in 2050, they assume the future board will use the equation indicated above substituting $latex dt = t-2050 $ (Note: if scientists uncertainty model improves, that will make the case for waiting better than presented here.)

   (c) With respect to decision to build in the start year (i.e. 2014) the current board will assume they will build $latex N_{build} $ times between now and 2100, each time building at the beginning of the period. So, the first build period begins in 2014. If there are two build periods, the second one begins 43 years from now in 2057. At each time, the board will apply the exact same method Lew used to estimate the proper wall height to build, except that rather than building to protect through 2100, they will build to protect up to the next scheduled build time. That is: in 2014, they will build to protect to 2057. Moreover, Lew actually provided three methods of estimating the added protection height required to permit flood risk in the final year match current flood risk , each using a different probability distribution function (pdf) for the uncertainty. Out of caution, the board will assume this pdf is gaussian which maximizes the predicted height of protections.

   Also, in it’s wish to use the exact same method as Lew, the current board will sift through his papers to find any parameters he used, and match those. They will notice he cites “Hunter” who uses a formula that includes a parameter $latex \lambda $ which is assumed constant with time. The board will read Lew’s paper, and discover this bit of text that contains numbers that permits them to back out the value of $latex \lambda $ used by Lew:

   When uncertaintySLR is non-zero, then irrespective of what assumptions are made about the distribution of SLR, the required protective response increases and deviates rapidly and
   in an accelerating manner from the anticipated mean SLR. For example, under a Gaussian assumption, if uncertaintySLR is around 0.36 m, this raises the required protective response to around 1m. That is, an expected SLR of 0.5m requires that dikes and levees be raised by twice that amount in order to keep the risk of flooding constant in light of uncertaintySLR. If other distributional assumptions are made, the values change but the in-principle conclusion remains the same:

   (Note: consultation with Hunter, suggests Hunter used the (5%-95%) uncertainty range and the 0.36 m may be a typo; Hunter shows 0.26m. I’ll be using the 0.36 m as my goal is to make a qualitative rather than quantitative point. )

   Note: text corresponds to a discussion of the point highlighted in the figure below:

   

   My future figures will use this value of $latex \lambda $ which is held constant, as Lew suggests.

   At future build times, the current board assumes the future board will act as follows: if based on the observations available at that future time the height of the wall is estimated to be sufficient to protect the village at the end of the next build time, the future board will skip that build but will not demolish it, otherwise, they will build to protect using the updated risk model that incorporated knowledge of the current observed sea level.

Results! (Otherwise known as the fun part!)

First: My results represent a detailed discussion of the height of the protections for the case represented by the highlighted point in “Lew’s” figure above. Recall the fundamental reason my results will differ from Lew’s is I permit the board to build in increments and in particular, they may build more than once. I will call the number of planned builds $latex N_{build} $. Since I am using Lew’s method of risk analysis, my results will reproduce his when the number of builds is $latex N_{build} =1 $ as that is the case his analysis represents. (Note however, I didn’t spend much time matching. I had to back a parameter out form the text comparison of two papers suggests a typo on one of the other. The results are somewhat sensitive to that parameter.)

To obtain results, I coded in the decision algorithm describe above in “R”. I did this because as far as the current board is concerned, the future observed values of sea level rise, $latex O[SLR](Y) $, are random. So, I made $latex O[SLR](Y) $ in build periods after the initial one to be a random variable with standard deviation equal to that estimated for the projected time between builds and mean equal to the updated projection for $latex E[SLR](Y) $. Recall that updated projection in future years is based on $latex O[SLR](Y) $ for the most recent observation. I also coded to implement the boards decision to only build if the expected value of the protections at the end of the current build period exceed that of the current wall. Results below are based on 10,000 iterations.

Expected Height of Protections
First, recall, in Lew’s analysis, there was 1 build. For the case I picked out, the “required” height of protections was 1-m with no variability in the height the board might build. Essentially, the board figures out the level they ‘need’ to build given current information, and build it. They are done. That result is represented by the black square above ‘1’ build below. Meanwhile, the blue circle represents the expected value of sea level rise $latex E[SLR] $.

Note that in the $latex N_{build} case, on average, the protection will exceed the amount that is actually required to meet acceptable flood risk by 0.5 meters in order to protect 2100 citizens adequately. This was the horrible idea that Lew’s text suggest is somehow “required”.

Next look at the figure, allowing your eyes to travel to the right. Suppose the board decides to build twice, $latex N_{build}=2 $, with an initial build now, and a second one 43 years from now. In this case, we can’t know the level of protections the board will judge proper 43 years from now. That magnitude will depend on the observed sea level rise. However, what we can examine is the expected value of the protections they will build, and its standard deviation. In this case, the expected value for the wall they will ultimately build 0.53m; this is substantially smaller than the 1-m they would build if they planned protecting citizens of 2100 using a protections built in 2014. Morever, it’s only a smidge above the expected sea level rise of 0.5m.

Looking further to the right, you can see the expected value of the protections declines as the number of build increments increases approaching 0.5 m as the number of builds increases to infinity. At this point, it is worth noting that qualitatively this result is fairly general: The expected value of the wall height required to match flood risk in the beginning and end periods will tend to diminish. However, the quantitative results are affected by functional form the board assumes for projections and its uncertainty.

Next: it’s worth admitting that the future boards may build protection levels that are either higher or lower that the best estimate for the future protection height. The ±90% spread is illustrated with the blue uncertainty bars. Generally speaking, under the multiple build scenario, boards will build for higher protection if sea level actually rises at a higher rate than anticipated currently and lower if lower if it rises at a lower rate. Interestingly, under the current set of assumptions for the parameters (both creating quadratics), the height of protections ‘required’ if the board builds only once lies outside the ±90% spread of heights they will build if they defer part of their decision for 43 years. So: the ‘build full barriers now‘ tends to result in over building, which is unnecessarily costly. (Necessary funds might need to be taken from lunch subsidies for low income children, or for medical care for the elderly. Who knows?)

But some might think: We’ll at least the public will get ‘better’ protection. Sort of. Recall that even if the height of the wall built becomes deterministic under Lew’s “1 build” scenario, the future sea level is a random variable. So the height of protection actually required in 2100 is a random variable whose mean is 0.5m and ±90% variability is 0.36m The following graph compares the height built to the height of protection actually required in 2100.

Notice that in the figure, the height of the wall and it’s ±90% uncertainty intervals are well away from 0m. This means that in more than 95% of future outcomes, the public has much more protection that required to maintain adequate flood risk. In fact, they would have obtained the level of protection the board thinks is adequate building a wall that is more than 0.14 m lower.

In the other cases, there is a possibility that when 2100 comes along, the wall is a bit too short. With two builds, when the final protection height is too short relative to the height required for adequate flood risk in 2100, the short fall is based on the sea level rising faster during the final period than anticipated based on the previously observed periods. In the case computed, there is 5 % chance the wall is 0.06m too short to give whatever level of protection the board deems adequate.

This too-short wall be seen as a big “disadvantage”, but fear not. The future board can schedule another build. If they believe the sea has stopped rising, they can add 6 cm to the wall elevation. If the 2100 board believes the sea is continuing to rise, and they believe the area continues to need protection, the future can base their decision on new, updated, hopefully improved projections of uncertainty.

Summary

1. There is no mathematical reason a current, 2014, board needs to build a protections to levels required to protect citizens in 2100.
2. There is no mathematical reason a current, 2014, board needs to build to protect to the upper uncertainty bound for sea level rise in 2100.
3. If the board opts to schedule several many periods, the best estimate for the required protected height approaches the mean value for the projected sea level rise.
4. If the board opts to build as required, they can come close to building “just the right” height protections.
5. Other factors not discussed here become very important to the boards decision. These include: the discount rate which makes current expenditures more costly than future ones, incremental added cost of maintaining unnecessarily tall protections for 100 years, risk of unnecessary excess loss if the unnecessarily tall protections are destroyed by an earthquake sometime between 2014 and the time when the flood protection of that height might be needed and added costs when engineering projects start and stop. Most of these will tend to argue in favor of many builds; the final one argues in favor of a smaller number of builds. Careful calculations would be required to determine the optimum number of builds; it is unlikely to be 1.
6. It is true that uncertainty results in higher costs. However, note for the case considered, the “Lew” method suggested the uncertainty meant one needed to build 1-m protections when the best estimate for required protections under certainty was 0.5 m. But by responding with sanity the expected value of build heights were (0.53 0.51 0.50 0.50)m for (2,3,4,5) builds respectively, with the additional height above the 0.5m required under certainty falling within rounding error. Admittedly, rounding down was required, but I think few boards would be impressed by the thought that ‘climate uncertainty’ adds horrific costs when the difference in cost is less than 0.5% of the expected costs, represents less than 1mm in height of a protection and this uncertainty is dwarfed by other uncertainties that affect board decisions.

Anyway, I thought some of you might enjoy this “toy math” post. I did.

Links to papers
Readers might want these handy links:
A simple technique for estimating an allowance for uncertain sea-level rise. John Hunter y analysis is an application of equation (6).

Scientific uncertainty and climate change: Part I.
Uncertainty and unabated emissions
Stephan Lewandowsky · James S. Risbey ·
Michael Smithson · Ben R. Newell · John Hunter

This post is a follow on to my earlier post discussing what happens if we were to believe the uncertainty in the climate sensitivity rose relative to our current understanding while our best estimate of the expected value in temperature rise ($latex E[T] $) remained constant. For now, the focus will simply be numbers. For the purpose of this blog post, I make the same assumptions discussed in my previous post and assume the damage function (i.e. present value of costs ) for climate change is a function of the temperature rise due to doubling of CO2 and that it the costs in quatloos are a quadratic function of temperature with zero cost at no rise. That is $latex C(T) < T^{2} $ . This is the choice Lewandowsky made in his post on implications of uncertainty which touchted on costs. Some results I will discuss depend on the choice of damage function; others do not.

Some discussion of the economic model
I have selected the damage function of the form $latex C(T) < T^{2} $ where C is the present value of costs arising from damages from climate change denominated in Quatloos and T is the temperature rise under doubled CO2. I selected this form because it is the choice Lewandowsky made. This is a somewhat cartoonish choice is suitable for the current discussion which, despite containing numerical values, is qualitative. Even though this is a cartoon model, I think it's important to understand the features the cartoonish economic model is intended capture. One of these features is the effect of "temporal discounting". In Lewandowsky's post that touched on costs, there was quite a bit of huffing and puffing (some of it nonsense) about economists temporally discounting costs. For example:

This is an important point to bear in mind because if the greater damage were delayed, rather than accelerated, economists could claim that its absolute value should be temporally discounted (as all economic quantities typically are; see Anthoff et al., 2009). But if greater damage arrives sooner, then any discounting would only further exacerbate the basic message of the above figure: Greater uncertainty means greater real cost.

Economists do temporally discount costs. (In fact, most engineers take engineering economics to help guide rational choices about mundane things like spending money on improvement to physical plant.) One reason costs are temporally discounted is simple and is touched on in 8th grade home economics. Here’s an example on might discuss in 8th grade:

Suppose you have the opportunity to buy a refrigerator now. It costs $700. You have $700. You currently don’t need and can’t use this refrigerator but anticipate you will need a refrigerator a year from now when you move from your dorm to an apartment. You also know you can lend your money safely to an enterprising friend who will pay you back $770 a year from now and you are 100% certain they will pay you back. You are also 100% certain the refrigerator will cost $720 a year from now. Given this fact pattern, should you buy the refrigerator now or wait a year?

Written this way, everyone knows they should defer buying the refrigerator and lend the money to their enterprising friend. The $70 you make by not spending your money and putting it to work means you will easily cover the $20 cost increase in the price of the refrigerator. The “discount rate” is the formal method of accounting for the fact that available money can generally be used to do something that creates value. In economics, these valuable things are denominated in money; in this example, the extra value is $70.

Because this post is largely motivated by Lewandowsky’s discussion, it’s worth nothing that if we are going to do the sort of cartoon analysis he did and which I largely imitate here, the damage function (i.e. estimate of the present value of costs based on the change in climate sensitivity) has already been assumed to include the temporal element. That is the discount rate has already been applied.

This must be so because mathematically speaking the damage function we chose for this exercise is only a function of climate sensitivity and so must be understood depend only on the climate sensitivity. (More complicated models can be created from more detailed damage functions by using cost models that account for time, finding more complicated probability distribution functions for scenarios of temperature. From these one might develop the simpler more cartoonish model shown here.)

The consequence of the fact that the cost function used here already accounts for the time evolution means that one ought not to suggest the the cost climate change should be further inflated to account for the fact that they may arise sooner rather than later because that effect is already accounted for in the cost model. When any such suggestion follows an analysis using a cost model of this sort, it can be taken as evidence the person making the suggestion may not understand the math underlying their cartoon analysis. Alternatively, they have forgotten the implications of their earlier assumptions.

Results of effects of uncertainty on estimated costs of climate change
Now, based on the cost model described above, I’m going to present some simple graphics showing what happens to our estimate of the probable costs of climate change if we use the sort of cost model used by Lewandowsky. The analysis is simple: To compute the probablity density function for costs I inverted my cost function to obtain the temperature rise as a function of cost, inserted that into the pdf for temperature (shown in the previous post) and used the change rule to change dCost = dT (∂C/∂T). I coded that up, and did higher mathematics also called “summing things up” (i.e. integrated.)

The results in graphical form are below. On the left, I show the cumulative probability distribution of costs computed in the ‘base case’ (i.e. lower variance.) To the right I show the values computed assuming that the best estimates of temperature is fixed but the standard deviation increases. I did this by increasing the standard deviation of our uncertainty in feedback parameter from Roe and Baker (2007).

Examining the figure we can see some features that are independent of the cost model; others are dependent on the cost model.

The feature that is independent (or nearly independent of) the choice of economic model.

1. Increases uncertainty in our ability to predict the cost. That is: the range of costs we consider possible increases. This is tautological.

Other features we predict that depend our choice of economic model. (It is worth nothing some depend on the shape of the probability distribution for the uncertainty in our ability to predict the feedback parameter.) If the expected value of the rise in temperature was correctly reflected in our first analysis, but the standard deviation our “true” uncertainty in feedback parameter is twice than value we used to compute costs:

1. In both the base case and the higher uncertainty case, the expected value of the cost is greater than the cost computed based on the expected value of the temperature. That is $latex < E( C[T]) $. This will always occur when the present value of the costs are assumed to be a quadratic function of the temperature, but might not occur for other economic models. It's worth nothing that this observation doesn't need we need to "worry more". It merely means that when estimating costs, one uses $latex E( C[T]) $ not $latex C(E[T]) $. The principle that one uses $latex E( C[T]) $ is well established in economics would hold even if the inequality reversed. For this reason, reports generally mention $latex E( C[T]) $.
2. In the base case, the expected cost of unmitigated climate change ( $latex E( C[T]) $ ) is 13.9 Quatloos. What this means is that under the base case, if we had only two choices: a) do nothing or b) undertake a perfect method to prevent temperature from rising at all, we should undertake that method provided the present value of its cost is less than 13.9 Quatloos. Otherwise, if we believe our cost model properly accounts for the costs at all levels of climate sensitivity and our statistical model describing the uncertainty in climate sensitivity is correct, we should select “do nothing”. That’s how economic models are used.
3. In the base case, the probability that costs would fall below 13.9 Q is 62.7%. That means that if we do spend 13.9 Quatloos to avoid climate change, there is a 37.3% chance we will have spent more Quatloos than we ought to have. How much we would have wasted will depend on the actual temperature rise– which we know we cannot predict. (Note: if we spend the 13.9 Quatloos and the method works, we may never know the actual costs that would have occurred if we did not spend the Quatloos.)

   In contrast, there is a 37.3% changes damages will exceed 13.9 Q if we do nothing. The damages may possibly exceed 13.9 Q by a large amount. That means that it is possible that damages could be some amount– say 35 Quatloos. If we are presented with a prefect mitigation strategy whose cost has a present value of 14 Q, and we pass that up because it costs more than 13.9 Q, we will end up being out of pocket 35 Quatloos when we could have gotten away with spending only 13.9 Quatloos.

   Note however, that the skewed distribution means that if we use expected value of costs to decide whether to undertake a mitigation project we are more likely to spend too much rather than spend too little The flip side is that in cases where we spend too little, our costs might be very large. It is generally accepted that using the expected value balances these two issues properly.

4. The estimate for expected value of costs ( $latex E( C[T]) $ ) computed using by doubling our estimate of the uncertainty in the standard deviation of the climate feed back is is 17% higher than anticipated using the base case. That means that means that if we estimate using the higher level of uncertainty, we should chose mitigation strategies even if they cost a 17% more than under the base case.
5. Even though the the expected value of the cost of climate change increased to 16.3 Quatloos when we considered the possibility that the uncertainty in our estimate of feedback parameter, the probability that the cost will exceed 13.9 Q we fall below the value we estimated in the basecase is 64%. That is: We are still more likely to spend a little too much. That said: the item we should focus on ins the expected value of the cost: 16.3 Quatloos.
6. For those who like to consider the extreme outcomes: Whereas in the base case, we anticipate there is a 90% chance the costs will fall between 3.7 Quatloos and 33.23 Quatloos, we now think the 90% range is 1.6 Quatloos to 53.6 Quatloos. Expressed in Quatloos, the width of the interval encompassing 90% of the likely outcomes increased by 76%.
7. Comparing the cost estimate under higher uncertainty to that under base case (i.e. lower uncertainty), likelihood that costs will be lower than anticipated lower bound based on the base case increases from 5% to 19.8%. The likelihood that costs will be higher than the anticipated upper bound computed in the base case increases from 5% to 12.1%. So, the likelihood that costs will fall below the level we anticipated under the base case rises more rapidly than the likelihood costs will rise above the level we anticipated.

It seems to me those are the major observations one can make about how uncertainty affects costs under my assumptions about the probability distribution function for our feedback parameter using the simple cost function and using feedback parameter values from one of the cases described in Roe and Baker. Of course as with all results involving statistical models, assumptions made prior to doing any math matter. Numerical values would differ if we chose a different cost function a different probability distribution function and so on. It’s perfectly legitimate to suggest the cartoon cost function is not correct (it’s not) or that one might pick a different probability distribution function for feedback. I certainly invite debate on those issues– my goal is to show a cartoon or ‘toy’ analysis that is food for thought.

So, what should we do?
A bit later on, I’ll write a post discussing how we would use the results of analysis of this sort (whether cartoon or real) to make decisions about mitigation strategies. Much of that will be plain old discussion with no graphs or math but I’ll try to avoid rambling about how much we should “worry” or whether or not uncertainty is our “friend”. Naturally, if the analysis involves making decisions based on cost, then the discussion will involve making decisions based on costs. As we can see above, uncertainty affects those costs— generally speaking greater uncertainty increases the expected values of costs of everything. This includes both the cost of the impacts of climate change and the cost of mitigation strategies.